According to the FHS any files within /var/cache should be able to be removed without issue. In practice, some apps will not like it, but you should be good with TKLBAM. Please note though that it may re-download some/all of that next time an incremental backup runs (it uses the cached files to work out what has changed when calculating the incremental backups).

The /mnt/tmp dir is a temporary directory which can also be safely deleted. Although assuming that you are using TKLBAM for regular backups, personally, I'd be inclined to run a full backup manually afterwards to double check that everything is working as it should (if the last backup was interrupted, it may be expecting to resume next time it runs and may fail if the temp & cache files have been removed). This should do the trick:

tklbam-backup --full-backup now

If it errors because of a "file not found" error or similar, then try using the --disable-resume switch as well. I.e.:

tklbam-backup --disable-resume --full-backup now

If you are concerned about the size of your backup, then you can tune your backup somewhat as noted in the faq/docs. Also if you have some fairly static files and some files that change more rapidly, it's also possible to separate them into different backup sets (i.e. multiple backups on a single machine) if you wish.

Hope that helps.